A FEniCS Tutorial - FEniCS Project

Instead of nabla_grad we could also just have written grad in the examples
in this tutorial. However, when taking gradients of vector fields, grad and
nabla_grad differ. The latter is consistent with the tensor algebra commonly
used to derive vector and tensor PDEs, where the ”nabla” acts as a vector
operator, and therefore this author prefers to always use nabla_grad.
HavingaandLdefined,andinformationaboutessential(Dirichlet)boundary
conditions in bc, we can compute the solution, a finite element function u, by
u = Function(V)
solve(a == L, u, bc)
Some prefer to replace a and L by an equation variable, which is accomplished
by this equivalent code:
equation = inner(nabla_grad(u), nabla_grad(v))*dx == f*v*dx
u = Function(V)
solve(equation, u, bc)
Note that we first defined the variable u as a TrialFunction and used it
to represent the unknown in the form a. Thereafter, we redefined u to be a
Function object representing the solution, i.e., the computed finite element
function u. This redefinition of the variable u is possible in Python and often
done in FEniCS applications. The two types of objects that u refers to are
equal from a mathematical point of view, and hence it is natural to use the
same variable name for both objects. In a program, however, TrialFunction
objects must always be used for the unknowns in the problem specification (the
form a), while Function objects must be used for quantities that are computed
(known).
The simplest way of quickly looking at u and the mesh is to say
plot(u)
plot(mesh)
interactive()
The interactive() call is necessary for the plot to remain on the screen. With
the left, middle, and right mouse buttons you can rotate, translate, and zoom
(respectively) the plotted surface to better examine what the solution looks like.
Figures 1 and 2 display the resulting u function and the finite element mesh,
respectively.
It is also possible to dump the computed solution to file, e.g., in the VTK
format:
file = File(’poisson.pvd’)
file